Dividing a polynomial by a binomial
Divide polynomials by binomials in Grade 9 algebra by factoring the numerator when possible and canceling the binomial factor, or using polynomial long division when factoring is not straightforward.
Key Concepts
Property If possible, factor the numerator and denominator, then divide out any common factors to simplify the expression. $$ \frac{(x+a)(x+b)}{(x+a)} = x+b $$ Explanation This is the ultimate shortcut for division! It’s like finding matching socks in a messy drawer. Once you spot a matching factor on the top and bottom, you can cancel them out. What’s left over is your much tidier, simplified answer. Easy peasy! Examples $$ (x^2 + 8x + 15) \div (x+3) = \frac{(x+5)(x+3)}{x+3} = x+5 $$ $$ \frac{x^2 6x + 9}{x 3} = \frac{(x 3)(x 3)}{x 3} = x 3 $$.
Common Questions
What is the simplest method to divide a polynomial by a binomial?
First try to factor the numerator. If the binomial divisor is a factor of the polynomial, cancel it. For (x² + 5x + 6) ÷ (x + 2), factor the numerator to (x+2)(x+3), cancel (x+2), leaving x + 3.
How does polynomial long division work?
Set up like numeric long division. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Multiply, subtract, bring down the next term, and repeat until the degree of the remainder is less than the divisor.
When does dividing a polynomial by a binomial produce a remainder?
A remainder occurs when the binomial is not a factor of the polynomial. For example, (x² + 5x + 7) ÷ (x + 2) leaves a remainder of 1, written as x + 3 + 1/(x+2). No remainder means the binomial divides evenly.